2012/11/07 by Jih-Hsin Cheng, Cheng, Jih-Hsin, Jenn-Fang Hwang +1
Computer Science · Mathematics · #32V20 #35J70 #35L80 #49Q10 #53A10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1211.1474
openalex publication_date 2012/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the uniqueness of generalized p-minimal surfaces in the Heisenberg group. The generalized p-area of a graph defined by u reads ∫ |∇ u+F| + Hu. If u and v are two minimizers for the generalized p-area satisfying the same Dirichlet boundary condition, then we can only get N_F(u) = N_F(v) (on the nonsingular set) where N_F(w) := \frac∇ w+F|∇ w+F|. To conclude u = v (or ∇ u = ∇ v), it is not straightforward as in the Riemannian case, but requires some special argument in general. In this paper, for a generalized area functional including p-area, we prove that N_F(u) = N_F(v) implies ∇ u = ∇ v in dimension ≥ 3 under some rank condition on derivatives of F or the nonintegrability condition of contact form associated to u or v. Note that in dimension 2 (n=1), the above statement is no longer true. Inspired by an equation for the horizontal normal N_F(u), we study the integrability for a unit vector to be the horizontal normal of a graph. We find a Codazzi-like equation together with this equation to form an integrability condition.